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Categories for the Working Mathematician (Graduate Texts in Mathematics) [Hardcover]

by Saunders Mac Lane
4.1 out of 5 stars  See all reviews (12 customer reviews)

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Book Description

October 1, 1998 0387984038 978-0387984032 2nd
An array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. It then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterised by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence.

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Categories for the Working Mathematician (Graduate Texts in Mathematics) + Category Theory (Oxford Logic Guides)
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Editorial Reviews


Second Edition

S.M. Lane

Categories for the Working Mathematician

"A very useful introduction to category theory."—INTERNATIONALE MATHEMATISCHE NACHRICHTEN

Product Details

  • Series: Graduate Texts in Mathematics (Book 5)
  • Hardcover: 317 pages
  • Publisher: Springer; 2nd edition (October 1, 1998)
  • Language: English
  • ISBN-10: 0387984038
  • ISBN-13: 978-0387984032
  • Product Dimensions: 9.5 x 6.3 x 0.9 inches
  • Shipping Weight: 1.4 pounds (View shipping rates and policies)
  • Average Customer Review: 4.1 out of 5 stars  See all reviews (12 customer reviews)
  • Amazon Best Sellers Rank: #377,911 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
49 of 51 people found the following review helpful
5.0 out of 5 stars One of the great books in mathematics September 24, 2000
This book is a classic. Clearly written, drawing on a vast number of different applications and motivations for the subject. Eilenberg and Mac Lane created category theory and this book is alive with the very style of thought Mac Lane brought to it in the first place. It is obvious that Mac Lane wrote each page, and each exercise, with a view of the whole book in mind. He starts with the very basics, assuming indeed that you know nothing of category theory. He goes on to adjunctions, limits, the adjoint functor theorems, monads (triples), monoidal categories, Abelian cateories, Kan extensions, higher dimensional categories, and categorical foundations. It is a masterpiece and one of the great books in mathematics.
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46 of 49 people found the following review helpful
5.0 out of 5 stars A Classic July 2, 2004
Format:Hardcover|Verified Purchase
Well, let us think about this a little bit...You want to learn Category theory, whether for some course or just for the fun of it, and now where do you turn in order to learn the necessary concepts. If you are a mathematician and have some experience, then you turn to the masters, the originators of the given subject and read their work. Sure, being the founder of a given subject does not imply that you are a good expositor and hence are capable of revealing the necessary concepts for the beginner-allow me to inform that Mac Lane is indeed as good as an expositor as he was a mathematician. For any doubters, I point you to the only other text you should read on Category theory, namely, "Category Theory" by Horst Herrlich and compare this text with Mac Lane's. Aside from that, and with respect to the text, for most beginners or interested readers I would suggest the following outline: Read 1.1-6; 2.1-3 & 8 possibly 2.4; all of 3; as for 4 skip section 3; 5.1-5; all of 8. Then, dependent upon your desires and or focus as well as your mathematical ability, it should become obvious which of the remaining topics should be read. Finally, the only other source I would recommend for learning Category theory can be found on-line using the keyword 'Awodey'. Anyways, Enjoy and good luck.
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38 of 43 people found the following review helpful
4.0 out of 5 stars Definitely a grad text July 22, 2001
By A Customer
This book is extraordinarily well written. It covers the necessary topics in a concise, orderly manner. HOWEVER, it presumes a substantial amount of knowledges concerning various algebraic/abstract structures in the field of mathematics. If you already have had experience with such structures, and are simply looking to understand them from a different perspective - this is the book for you. However, if you have limited knowledge with regards to advanced math (ie - grad level math) then try the book 'Arrows, Structures and Functors: The Categorical Imperative' by Manes and Arbib. This introduces the reader gradually to simple algebraic structures, monoids, groups, metric spaces, topological spaces, and the categories that can be built around them.
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17 of 19 people found the following review helpful
5.0 out of 5 stars Simply Great January 4, 2008
Have you ever tried reading Descartes' "Geometry"? It's not a good place to learn about coordinate geometry. I tried. This was almost 10 years ago, but I still remember it pretty well. Ok, so maybe the experience was even a bit traumatic.

Usually when someone works out a theory, it takes a fresh perspective (or two, or ... you get it) to really digest it, and come up with a reasonable way of teaching it to newcomers. It's less evident nowadays, with improved communications technology and such, but people aren't exactly turning to Grothendieck's expositions as their intro to his geometry either. Mac Lane is an exception.

This book seems completely inapproachable. The title is scary. The topic is scary. Open to a random page and try to judge its accessibility: scary. Well, here's the real story: you need to know algebra through modules, and it'd be nice if this algebra background introduced "universals" like abelianization or free modules in a way that involved the diagrams and the unique mappings you get from the given ones. If this stuff makes any sense, you can read this book. It's not that scary. If you're up to the challenge, you might even enjoy it. This is actually my favorite book.

Here's the approach that I feel worked well for me:

- gloss over the set-theoretic foundations at first. Make sure you know the proper class/set and large/small category distinctions, but don't dwell on them much.

- focus on the examples that are familiar, but read through the others too. Mac Lane uses tons of examples to suit a variety of backgrounds, and his presentation is so clear that the theory can often explain the examples.

- trust the author. It may seem like product or comma categories deserve fuller treatment with more motivation. No.
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12 of 16 people found the following review helpful
4.0 out of 5 stars Classic and worth it August 12, 2003
It is difficult to make understand what "is" category theory. Is it a foundational discipline? Is it a discipline studying homomorphisms between algebras? Is it nonsense? Well, in my opinion this book does not help in gaining this kind of understanding. But all the stuff I read which have been written with that purpose in mind did not have any success - perhaps because I am not a mathematician, or perhaps because some concepts in category theory are really too abstract for anyone to give "an intuition" of them (you still can with functors and natural transformations, but try with adjointness...). This said, I found the book wonderful: Every concept is presented neatly. I use it as a reference each time I want a clear and rigorous definition of a concept. Sometimes this rigour helped me in gaining the famous intuition behind the concept.
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Most Recent Customer Reviews
5.0 out of 5 stars It is a Great Book. (although some notions are old)
It is a Great Book and it is quite helpful to me. (although some notions are old) The book is brand new.
Published 5 months ago by Zhenchao Ge
5.0 out of 5 stars Great Book
This book is kind to the reader yet it does not over-explain the theory. It is comprehensive and current. .
Published 6 months ago by Rafael H Alpizar
5.0 out of 5 stars The greatest math book ever written
Here are the design patterns of mathematics. Unlike the computer science design patterns, these compose!

If you're a mathematician, you've already read this book. Read more
Published 9 months ago by Foo Bar
3.0 out of 5 stars A "tough read"
Category Theory shouldn't be this tough.
I was trained as an electrical engineer, so, my math is reasonably good.
However, I can't get through this book. Read more
Published 15 months ago by
2.0 out of 5 stars Poorly written standard text.
This book has everything you need, but it is written in an abstruse style in my opinion.
Published on December 7, 2007 by D. Spivak
3.0 out of 5 stars You may not need this unless you major in category theory.
I entirely agree with the reviewer Lucas Wilman.
As a book by the creator of category theory, it has extensively incorporated relevant items. Read more
Published on May 13, 2004
3.0 out of 5 stars Kind of Dull
I read this due to its odd title. It is fairly easy to understand. It assumes that you have very little previous knowledge of the subject. For me it just wasn't that useful. Read more
Published on July 4, 2000
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